235 lines
8.8 KiB
TeX
235 lines
8.8 KiB
TeX
\subsection{Slice knots and metabolic form}
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\begin{theorem}
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\label{the:sign_slice}
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If $K$ is slice,
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then $\sigma_K(t)
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= \sign ( (1 - t)S +(1 - \bar{t})S^T)$
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is zero except possibly of finitely many points and $\sigma_K(-1) = \sign(S + S^T) \neq 0$.
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\end{theorem}
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\begin{lemma}
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\label{lem:metabolic}
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If $V$ is a Hermitian matrix ($\bar{V} = V^T$), $V$ is of size $2n \times 2n$,
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$
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V = \begin{pmatrix}
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0 & A \\
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\bar{A}^T & B
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\end{pmatrix}
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$ and $\det V \neq 0$ then $\sigma(V) = 0$.
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\end{lemma}
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\begin{definition}
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A Hermitian form $V$ is metabolic if $V$ has structure
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$\begin{pmatrix}
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0 & A\\
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\bar{A}^T & B
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\end{pmatrix}$ with half-dimensional null-space.
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\end{definition}
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\noindent
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Theorem \ref{the:sign_slice} can be also express as follow:
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non-degenerate metabolic hermitian form has vanishing signature.
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\begin{proof}
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\noindent
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We note that $\det(S + S^T) \neq 0$. Hence $\det ( (1 - t) S + (1 - \bar{t})S^T)$ is not identically zero on $S^1$, so it is non-zero except possibly at finitely many points. We apply the Lemma \ref{lem:metabolic}.
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\\
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Let $t \in S^1 \setminus \{1\}$.
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Then:
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\begin{align*}
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\det((1 - t) S + (1 - \bar{t}) S^T) =&
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\det((1 - t) S + (t\bar{t} - \bar{t}) S^T) =\\
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&\det((1 - t) (S - \bar{t} - S^T)) =
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\det((1 -t)(S - \bar{t} S^T)).
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\end{align*}
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As $\det (S + S^T) \neq 0$, so $S - \bar{t}S^T \neq 0$.
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\end{proof}
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\begin{corollary}
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If $K \sim K^\prime$ then for all but finitely many $t \in S^1 \setminus \{1\}: \sigma_K(t) = -\sigma_{K^\prime}(t)$.
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\end{corollary}
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\begin{proof}
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If $ K \sim K^\prime$ then $K \# K^\prime$ is slice.
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\[
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\sigma_{-K^\prime}(t) = -\sigma_{K^\prime}(t)
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\]
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The signature gives a homomorphism from the concordance group to $\mathbb{Z}$.
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Remark: if $t \in S^1$ is not algebraic over $\mathbb{Z}$, then $\sigma_K(t) \neq 0$
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(we can use the argument that $\mathscr{C} \longrightarrow \mathbb{Z}$ as well).
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\end{proof}
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\subsection{Four genus}
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\begin{figure}[h]
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\fontsize{20}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{0.7\textwidth}{!}{\input{images/genus_2_bordism.pdf_tex}}
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}
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\caption{$K$ and $K^\prime$ are connected by a genus $g$ surface.}\label{fig:genus_2_bordism}
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\end{figure}
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\begin{proposition}[Kawauchi inequality]
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If there exists a genus $g$ surface as in Figure \ref{fig:genus_2_bordism}
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then for almost all
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$t \in S^1 \setminus \{1\}$ we have
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$\vert
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\sigma_K(t) - \sigma_{K^\prime}(t)
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\vert \leq 2 g$.
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\end{proposition}
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% Kawauchi Chapter 12 ???
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% Borodzik 2010 Morse theory for plane algebraic curves
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\begin{lemma}
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If $K$ bounds a genus $g$ surface $X \in B^4$ and $S$ is a Seifert form then ${S \in M_{2n \times 2n}}$ has a block structure $\begin{pmatrix}
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0 & A\\
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B & C
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\end{pmatrix}$, where $0$ is $(n - g) \times (n - g)$ submatrix.
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\end{lemma}
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\begin{proof}
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\begin{figure}[h]
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\fontsize{20}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{0.5\textwidth}{!}{\input{images/genus_bordism_zeros.pdf_tex}}
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}
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\caption{There exists a $3$ - manifold $\Omega$ such that $\partial \Omega = X \cup \Sigma$.}\label{fig:omega_in_B_4}
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\end{figure}
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\noindent
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Let $K$ be a knot and $\Sigma$ its Seifert surface as in Figure \ref{fig:omega_in_B_4}.
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There exists a $3$ - submanifold
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$\Omega$ such that
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$\partial \Omega = Y = X \cup \Sigma$
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(by Thom-Pontryagin construction).
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If $\alpha, \beta \in \ker (H_1(\Sigma) \longrightarrow H_1(\Omega))$,
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then ${\Lk(\alpha, \beta^+) = 0}$. Now we have to determine the size of the kernel. We know that
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${\dim H_1(\Sigma) = 2 n}$. When we glue $\Sigma$ (genus $n$) and $X$ (genus $g$) along a circle we get a surface of genus $n + g$. Therefore $\dim H_1 (Y) = 2 n + 2 g$. Then:
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\[
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\dim (\ker (H_1(Y) \longrightarrow H_1(\Omega)) = n + g.
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\]
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So we have $H_1(W)$ of dimension
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$2 n + 2 g$
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- the image of $H_1(Y)$
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with a subspace
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corresponding to the image of $H_1(\Sigma)$ with dimension $2 n$ and a subspace corresponding to the kernel
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of $H_1(Y) \longrightarrow H_1(\Omega)$ of size $n + g$.
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We consider minimal possible intersection of this subspaces that corresponds to the kernel of the composition $H_1(\Sigma) \longrightarrow H_1(Y) \longrightarrow H_1(\Omega)$. As the first map is injective, elements of the kernel of the composition have to be in the kernel of the second map.
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So we can calculate:
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\[
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\dim \ker (H_1(\Sigma) \longrightarrow H_1(\Omega)) = 2 n + n + g -2 n - 2 g = n - g.
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\]
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\end{proof}
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\begin{corollary}
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If $t$ is not a root of
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$\det (tS - S^T) $, then
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$\vert \sigma_K(t) \vert \leq 2g$.
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\end{corollary}
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\begin{fact}
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If there exists cobordism of genus $g$ between $K$ and $K^\prime$ like shown in Figure \ref{fig:proof_for_bound_disk}, then $K \# -K^\prime$ bounds a surface of genus $g$ in $B^4$.
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\end{fact}
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\begin{figure}[H]
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\fontsize{20}{10}\selectfont
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\centering{
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\def\svgwidth{\linewidth}
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\resizebox{0.7\textwidth}{!}{\input{images/genus_bordism_proof.pdf_tex}}
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}
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\caption{If $K$ and $K^\prime$ are connected by a genus $g$ surface, then $K \# -K^\prime$ bounds a genus $g$ surface.}\label{fig:proof_for_bound_disk}
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\end{figure}
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\begin{definition}
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The (smooth) four genus $g_4(K)$ is the minimal genus of the surface $\Sigma \in B^4$ such that $\Sigma$ is compact, orientable and $\partial \Sigma = K$.
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\end{definition}
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\noindent
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Remarks:
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\begin{enumerate}[label={(\arabic*)}]
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\item
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$3$ - genus is additive under taking connected sum, but $4$ - genus is not,
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\item
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for any knot $K$ we have $g_4(K) \leq g_3(K)$.
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\end{enumerate}
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\begin{example}
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\begin{itemize}
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\item Let $K = T(2, 3)$. $\sigma(K) = -2$, therefore $T(2, 3)$ isn't a slice knot.
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\item Let $K$ be a trefoil and $K^\prime$ a mirror of a trefoil. $g_4(K^\prime) = 1$, but $g_4(K \# K^\prime) = 0$, so we see that $4$-genus isn't additive,
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\item
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the equality:
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\[
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g_4(T(p, q) ) = \frac{1}{2} (p - 1) (g -1)
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\]
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was conjecture in the '70 and proved by P. Kronheimer and T. Mrówka (1994).
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% OZSVATH-SZABO AND RASMUSSEN
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\end{itemize}
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\end{example}
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\begin{proposition}
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$g_4 (T(p, q) \# -T(r, s))$ is in general hopelessly unknown.
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\end{proposition}
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\begin{proposition}
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Supremum of the signature function of the knot is bounded almost everywhere by two times $4$ - genus:
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\[
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\ess \sup \vert \sigma_K(t) \vert \leq 2 g_4(K).
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\]
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\end{proposition}
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\subsection{Topological genus}
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\begin{definition}
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A knot $K$ is called topologically slice if $K$ bounds a topological locally flat disc in $B^4$ (i.e. the disk has tubular neighbourhood).
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\end{definition}
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\begin{theorem}[Freedman, '82]
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If $\Delta_K(t) = 1$, then $K$ is topologically slice (but not necessarily smoothly slice).
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\end{theorem}
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\begin{theorem}[Powell, 2015]
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If $K$ is genus $g$
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(topologically flat)
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cobordant to $K^\prime$,
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then
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\[
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\vert \sigma_K(t) - \sigma_{K^\prime}(t) \vert \leq 2 g
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\]
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if $g_4^{\mytop}(K) \geq \ess \sup \vert \sigma_K(t) \vert$.
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\end{theorem}
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\noindent
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The proof for smooth category was based on following equality:
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\[
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\dim \ker (H_1 (Y) \longrightarrow H_1(\Omega)) = \frac{1}{2} \dim H_1(Y).
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\]
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For this equality we assumed that there exists a $3$ - dimensional manifold $\Omega$ (as shown in Figure \ref{fig:omega_in_B_4}) which was guaranteed by Pontryagin-Thom Construction.\\
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Pontryagin-Thom Construction relays on taking $\Omega$ as preimage of regular value:
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\[
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H^1 (B^4 \setminus Y, \mathbb{Z}) = [B^4 \setminus Y, S^1],
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\]
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what relies on Sard's theorem, that the set of regular values has positive measure. But Sard's theorem doesn't work for topologically locally flat category. So there was a gap in the proof for topological locally flat category - the existence of $\Omega$.\\
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\noindent
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Remark: unless $p=2$ or $p = 3 \wedge q = 4$:
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\[
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g_4^{\mytop} (T(p, q)) < q_4(T(p, q)).
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\]
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% Wilczyński '93
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%Feller 2014
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%Baoder 2017
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%Lemark
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\\
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\noindent
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From the category of cobordant knots (or topologically cobordant knots) there exists a map to $\mathbb{Z}$ given by signature function. To any element $K$ we can associate a form
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\[
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(1 - t)S + (1 - \bar{t})S^T) \in W(\mathbb{Z}[t, t^{-1}]).
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\] This association is not well define because id depends on the choice of Seifert form. However, different choices lead ever to congruent forms ($S \mapsto CSC^T$) or induced the change on the form by adding or subtracting a hyperbolic element.
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\begin{definition}
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The Witt group $W$ of $\mathbb{Z}[t, t^{-1}]$ elements are classes of non-degenerate
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forms over $\mathbb{Z}[t, t^{-1}]$ under the equivalence relation $V \sim W$ if $V \oplus - W$ is metabolic.
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\end{definition}
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\noindent
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If $S$ differs from $S^\prime$ by a row extension, then
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$(1 - t) S + (1 - \bar{t}^{-1}) S^T$ is Witt equivalence to $(1 - t) S^\prime + (1 - t^{-1})S^T$.
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\\
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\noindent
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A form is meant as hermitian with respect to this involution: $A^T = A: (a, b) = \bar{(a, b)}$.
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\\
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$
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W(\mathbb{Z}_p) = \mathbb{Z}_2 \oplus
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\mathbb{Z}_2$ or
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$\mathbb{Z}_4$
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\\
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???????????????????????
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\\
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$\sum a_gt^j \longrightarrow \sum a_g t^{-1}$\\
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\begin{theorem}[Levine '68]
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\[
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W(\mathbb{Z}[t^{\pm 1}])
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\longrightarrow \mathbb{Z}_2^\infty \oplus
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\mathbb{Z}_4^\infty \oplus
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\mathbb{Z}
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\]
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\end{theorem}
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